Unidirectional amplification with acoustic non-Hermitian space time varying metamaterial - Institut Langevin
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ARTICLE https://doi.org/10.1038/s42005-021-00790-2 OPEN Unidirectional amplification with acoustic non- Hermitian space−time varying metamaterial Xinhua Wen1, Xinghong Zhu1, Alvin Fan1, Wing Yim Tam1, Jie Zhu2, Hong Wei Wu1, Fabrice Lemoult3, Mathias Fink3 ✉ & Jensen Li1 ✉ Space−time modulated metamaterials support extraordinary rich applications, such as parametric amplification, frequency conversion, and non-reciprocal transmission. The non- Hermitian space−time varying systems combining non-Hermiticity and space−time varying capability, have been proposed to realize wave control like unidirectional amplification, while 1234567890():,; its experimental realization still remains a challenge. Here, based on metamaterials with software-defined impulse responses, we experimentally demonstrate non-Hermitian space −time varying metamaterials in which the material gain and loss can be dynamically con- trolled and balanced in the time domain instead of spatial domain, allowing us to suppress scattering at the incident frequency and to increase the efficiency of frequency conversion at the same time. An additional modulation phase delay between different meta-atoms results in unidirectional amplification in frequency conversion. The realization of non-Hermitian space−time varying metamaterials will offer further opportunities in studying non-Hermitian topological physics in dynamic and nonreciprocal systems. 1 Department of Physics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China. 2 Department of Mechanical Engineering, the Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, China. 3 Institut Langevin, ESPCI Paris, PSL University, CNRS, 1 rue Jussieu, 75005 Paris, France. ✉email: mathias.fink@espci.fr; jensenli@ust.hk COMMUNICATIONS PHYSICS | (2022)5:18 | https://doi.org/10.1038/s42005-021-00790-2 | www.nature.com/commsphys 1
ARTICLE COMMUNICATIONS PHYSICS | https://doi.org/10.1038/s42005-021-00790-2 I n the past two decades, metamaterials with spatial modulation realize non-Hermitian space−time varying metamaterials by (with spatially varying parameters) have opened an emerging applying a non-Hermitian (gain and loss) space−time modula- paradigm to manipulate classical waves, and have been used to tion. In such a non-Hermitian space−time varying metamaterial, achieve many intriguing physical phenomena like negative the material gain and loss can be dynamically modulated and refraction, lensing, hologram, and invisibility cloaking1–6. To balanced in the temporal domain instead of the spatial domain. It endow metamaterials more tunability, numerous efforts have generates multiple harmonics while suppresses scattering at the been made to construct reconfigurable metamaterials whose incident frequency due to the elimination of the corresponding material parameters can be changed adiabatically from one set to Fourier component. Consequently, efficient frequency conversion another set, such as coding metamaterials and electronically- to the first-order harmonics can be achieved and unidirectional controlled active metamaterials, to realize tunable lensing, beam- amplification in frequency conversion with amplification ratio up steering, and topological edge states7–15. Temporal modulation of to 151% in signal amplitude is experimentally demonstrated. material parameters has been applied on metamaterials to further Moreover, due to the flexibility of our software-defined meta- extend the degrees of freedom in wave control16–30, which also material platform, we can arbitrarily define the “energy-level” opens further possibilities for extraordinary physics like tunable diagrams to manipulate the frequency-converted modes, in ana- frequency conversion18,19, non-reciprocal propagation20–29, time logy to quantum interference effect. reversal mirror31,32, parametric amplification33,34, and topologi- cal pumping35–37. Furthermore, analog concepts originally defined for spatially inhomogeneous materials can be explored in Results and discussion the temporal domain now, including the concept of temporal Non-Hermitian space−time varying metamaterial platform. effective media38,39, and time-gradient metasurfaces40, as well as Recently, research works on breaking time-reversal symmetry to the analogy of bandgap concept in temporal domain for fre- achieve temporal modulation of system parameters is a growing quency filtering41. area in both acoustics and optics. For example, to realize non- Recently, it has been proposed to add non-Hermiticity into reciprocal acoustic transmission, a mechanical driver is used to time-varying systems to achieve phenomena such as unidirec- temporally modulate the volume or the physical boundary of tional amplification42,43, bidirectional invisibility34, tunable resonators21,35. However, the modulation frequency and strength exceptional points by modulation44, and nonreciprocal edge state induced by the mechanical driver may be restricted in practical propagating in non-Hermitian Floquet insulators45. In the non- applications, e.g., the modulation strength decreases with Hermitian space−time varying systems, unidirectional amplifi- increasing frequency21. To get more flexible control, here we use a cation, referring to signal amplification in only one incident software-defined metamaterial with digital feedback circuits to direction, can be achieved by temporally modulating the coupling experimentally realize non-Hermitian space−time modulation. terms (both the real and imaginary part) between two Figure 1a schematically shows our time-varying platform in a 1D resonators42 or modulating gain and loss in both space and acoustic waveguide. It consists of two meta-atoms (termed as time43. Specifically, in ref. 42, the non-Hermitian coupling terms atom n = 1,2) separated by a small distance ‘ (0.04 m), and each can be used to control not only the target frequency-converted of them is constructed by one microphone (Dn ) and one speaker modes, but also other higher-order harmonics, to realize a one- (Sn ) with a feedback between them through an external micro- way amplifier. However, the experimental realization of uni- controller (Adafruit ItsyBitsy M4 Express). For a single meta- directional amplification in these non-Hermitian time-varying atom, the microcontroller is instructed R to perform a modulated 0 systems still remains a challenge. digital convolution S ðt Þ ¼ 2πγðtδtÞ∂ t aðtÞy ðt ÞD ðt t 0 Þdt 0 with yðt Þ ¼ Here, we construct meta-atoms with software-defined impulse g sin 2πfres ðt δt Þ e for t>δt and 0 for t
COMMUNICATIONS PHYSICS | https://doi.org/10.1038/s42005-021-00790-2 ARTICLE includes a programmable time-delay and the time delay of all the two constant amplitudes a1 and a2 with modulation period T electronic components. Here, we purposely choose δt in satisfying (equivalently modulation frequency F ¼ 1=T) and duty cycle 2πfres δt π. Then, the feedback from the microphone to the ξ : 1 ξ between the two static configurations. Figure 1b shows speaker ofeach meta-atom implements a harmonic response with the modulation in alternating between a Lorentzian configuration S f ¼ Y f Dðf Þ where Yðf Þ can be approximated by the fol- with positive a1 and an anti-Lorentzian configuration with lowing resonating form: negative a2 . The metamaterial is therefore being modulated between a lossy and a gain configuration. As a result, the meta- igfres f atom not only constitutes a frequency-dispersive time-varying Y f ffi 2 ; ð1Þ 2 fres f þ iγ polarizability47,48, but also includes temporally controlled mate- rial gain and loss. Moreover, we can apply the same aðtÞ on the next atom (atom 2) but with an additional time delay Δt in the The corresponding transmission coefficient tf/tb in the modulation. We define a modulation phase delay between forward/backward direction and the reflection coefficient r adjacent atoms as Δϕ ¼ 2πΔt=T, ranging from 0 to 2π for a (common for both directions) can be found as full cycle delay. It generally describes a space−time modulation aðt ðn 1ÞΔt Þ at different atoms. As aðtÞ is chosen as a periodic tf f ¼ 1 þ r f eik0 ð f Þ‘=2 ; tb f ¼ 1 þ r f eik0 ð f Þ‘=2 ; modulation, one can decompose all the signals in terms of Fourier Yð f Þ ð2Þ series, e.g., Dn ðt Þ ¼ Σm Dnm ei2πfm t , aðt Þ ¼ Σm Am ei2πmFt , where r f ¼ ik0 ð f Þ‘=2 ; 1e Yð f Þ the m-th harmonics is defined by fm ¼ f þ mF, with integer m labeling the order of the harmonics and f being the reference with reference plane set at the center between the microphone frequency, taken as the incident waves. Then, there can be and speaker, k0 f ¼ 2πf =c being the free-space wavenumber frequency conversions between the different frequency harmonics and c being the sound speed in air. More details in deriving the at atom n by operation are given in Supplementary Notes 1, 2. We note that in the limit of small ‘ ! 0, passivity (condition of meta-atom ðnÞ ðnÞ igfres Amm0 fm iðmm0 Þðn1ÞΔϕ to be Snm ¼ Ym;m 0 f Dnm0 ; Ym;m 0 f ffi 2 e : lossy) means j1 þ r j2 þjr j2 ≤ 1, equivalently Im Yðf Þ=i ≥ 0, fres fm0 þ iγ 2 which can be satisfied when both g and γ are positive numbers. ð3Þ We call such a configuration Lorentzian, being approximately lossy and having a Lorentzian line-shape in Yðf Þ=i or rðf Þ=i The radiation from the speakers, together with the background around the resonating frequency, as schematically shown in incident waves represented by the Fourier components of the Fig. 1c. On the other hand, to achieve gain, i.e., power is being microphone signal Dð0Þ nm , becomes the local fields detected by the injected into the meta-atom through the power supply of the microphones: microcontroller, one might want to flip the sign of γ but the Dnm ¼ Dðnm 0Þ þ eik0 ðfm ÞLnn0 Sn0 m : ð4Þ envelope of the convolution kernel yðtÞ will keep increasing in time, stopping a digital implementation of the convolution to be where Lnn0 is the distance between the microphone of atom n finished within one signal sampling period (chosen as Ts = 138.7 from the speaker of atom n0 . The repeated indices in formula μs here). Here, we choose a negative value of g (equivalently imply summation. Then, combining Eqs. 3 and 4 gives the positive g but negative a multiplying to it), meeting the gain microphones signals by solving condition ImðYðf Þ=iÞ
ARTICLE COMMUNICATIONS PHYSICS | https://doi.org/10.1038/s42005-021-00790-2 reflected signal. From Eqs. 3 to 5, we can see that the frequency such a configuration of two meta-atoms, captured in terms of an conversion is mainly enabled by the Fourier component of the “energy-level” diagram for two meta-atoms with the same aðtÞ as the term Amm0 for a conversion from m0 -th harmonics to resonating frequency (1 kHz) and modulation frequency m-th harmonics. The signal at the incident frequency is (90 Hz). The yellow bars describe the levels with resonance, and controlled by the direct current (DC) component: A0 . We note the levels with black color denote those allowed levels satisfying that in the above harmonic model for simplicity, we have omitted frequency transition rule: fout ¼ fin þ mF. The thin arrows the broadband response of the speaker and microphone, indicate the allowed frequency transitions in the presence of effectively lumped into g as a frequency-independent constant resonating mode, starting from a resonating level (yellow bar) to as approximation. The harmonic model and time-domain the converted levels (black bars). In the current case, we only simulation model with frequency dispersive speaker and micro- focus on the m = 1 and m = −1 transitions by fixing the incident phone response are detailed in Supplementary Notes 1−3 while wave at 1 kHz (thick red arrow). Different from the case of a the above harmonic model is presented in brevity. single meta-atom, there can be more than one path to get a specific frequency conversion performed. Take the harmonics of m = −1 as an example, the incident wave at 1 kHz resonantly Unidirectional amplification. With the reference static meta-atom, excites atom 1 and has a frequency transition to 910 Hz by a first we apply an amplitude modulation aðtÞ to obtain a time-varying order down conversion (red arrow). On the other hand, the meta-atom. Figure 2a shows the experimental results of forward incident wave equivalently excites atom 2 and have the same reflection spectrum rf =i (revealing polarizability of a meta-atom) frequency down conversion (gray arrow). These two paths can along the 1D waveguide for three amplitude-modulated cases now undergo interference with each other. (symbols) with fixed duty-cycle ξ ¼ 0:5 and modulation frequency The modulation phase delay Δϕ between the two meta-atoms F ¼ 90 Hz. The first case is a square-wave modulation with a1 ¼ 1 provides us a flexible knob to control such an analogy of quantum and a2 ¼ 0:6 on the meta-atom, and the measured reflection spectra interference. Figure 3b shows the experimentally extracted forward rf =i are plotted as symbols in black color in Fig. 2a. Here, the reflection amplitude rf , the reflection of the harmonic at incident measurement is done on the same frequency as the incident fre- frequency (fout ¼ fin ¼ 1 kHz) remains low in magnitude due to the quency (fout ¼ fin ). Both the real and imaginary part of reflection gain-loss balance in temporal domain, and the amplitude of spectrum for such a time-varying meta-atom shows a Lorentzian line converted harmonics (for both m = −1 and m = 1) varies with Δϕ, shape and match well with the reflection spectrum for a static meta- which are also shown in details in Fig. 3c and 3d respectively. atom (black line) with an effective amplitude a ¼ ða1 þ a2 Þ=2 ¼ 0:8 Taking m = 1 as example (1000 Hz ! 1090 Hz, Fig. 3d), the (black line). Such an average effect has also been verified in the same experiment results (symbols) match well with the numerical results figure for the case involved the material gain: an amplitude mod- based on the harmonic model (lines) (see details in Supplementary ulation with a1 ¼ 1:4 and a2 ¼ 0:6 (red symbols) to give an Note 3). The interference dips indicated by arrows in Fig. 3d are effective strength a ¼ 0:4 (red line). Such an equivalence can be obtained from the destructive interference condition (see details in derived from the harmonic model (Eqs. 3−5) with the approxima- Supplementary Note 5): tion that if the incident frequency is around resonance, the frequency-converted signal has only a small chance by another fre- quency conversion back to the incident frequency. Then, the har- 2πfin ‘ 2πfout ‘ ± Δϕ þ þ ¼ ð2n þ 1Þπ; n 2 Z ð6Þ monic response at the incidence frequency is dominated by A0 ¼ a. c c Based on this effect, we can obtain a time-varying meta-atom with nearly zero reflection spectrum at the incident frequency (blue where þðÞ sign indicates for the case of forward (backward) symbols) by applying an amplitude modulation with a1 ¼ 1 and a2 ¼ 1 (effective strength a ¼ 0: zero DC value of the modula- incidence. At Δϕ ¼ 1:4π where the measured backward reflection tion). In such a case, the reflection amplitude at the m-th harmonics amplitude rb ¼ 0:06 is minimum (destructive interference), the approximately come from the Ym;0 or the Am term in Eq. 3. measured forward reflection amplitude can actually achieve a Figure 2b shows the whole spectrum for such a time-varying maximum at rf ¼ 1:51 (constructive interference), giving us a meta-atom for fin from 750 Hz to 1250 Hz while fout is measured unidirectional amplification phenomenon. Huge contrast in reflec- from 650 Hz to 1350 Hz. It clearly shows a diminished the tion amplitude is also found in other phenomena like unidirectional harmonic at incident frequency (m ¼ 0) due to the temporally invisibility and non-Hermitian exceptional point49,50, while the balanced gain and loss. In addition, the harmonics of m ¼ ± 1; 3 combination of non-Hermiticity and time-varying capability allows generated by frequency conversion fout ¼ fin þ mF (dashed lines us to generalize it to frequency-converted signals and the contrast labeling the different m) show significant reflectance amplitude. It can be even made larger with amplification on one side incidence. is observed that the 1-st order harmonics (m ¼ ± 1) are more The demonstrated unidirectional amplification comes from the prominent, and the presence of resonance greatly enhances the ability of having gain in one direction but not the other direction. In conversion efficiency (reaching around 60% near the resonating fact, the system also needs to work in the regime of stability, which frequency at 1 kHz). As shown in Fig. 2c, when duty cycle ξ varies is particularly important when we have gain in the system. For the from 0 to 1 for incident frequency at 1 kHz while the modulation current case of two time-varying meta-atoms, the stability condition frequency is still kept at constant 90 Hz, the gain-loss balance can be obtained by requesting all the poles of the system response only happens at ξ ¼ 0:5 to get diminished at the 0-th order matrix from Eq. 5, equivalently the zeros of det I T ðf Þ ¼ 0, harmonics (fout ¼ fin ¼ 1 kHz) and most prominent converted lying on the lower half of the complex frequency plane. Detailed harmonics (see more gain-loss balance case in Supplementary theoretical analysis (in Supplementary Note 6) confirms the stability Note 4). At ξ ¼ 0 (ξ ¼ 1), the atom goes back to the static atom of the system, until we increase the resonating strength of the meta- with gain (loss), no frequency conversion occurs as expected. We atoms to at least 4.8 times of the current value. In reality, the note that the up-converted mode has a larger amplitude than the frequency-dispersive speaker and microphone responses complicate down-converted mode due to the frequency dispersive response the analysis while experimentally, the resonating strength can be of the speaker and detector (see Supplementary Note 1). increased to around 5.7 times of the current value until sound in With such time-varying meta-atoms, we now design the non- audible level is observed with no incident waves to indicate the Hermitian space−time varying metamaterials. Figure 3a shows stability threshold. 4 COMMUNICATIONS PHYSICS | (2022)5:18 | https://doi.org/10.1038/s42005-021-00790-2 | www.nature.com/commsphys
COMMUNICATIONS PHYSICS | https://doi.org/10.1038/s42005-021-00790-2 ARTICLE (a) 1090 Hz (c) 0.8 fout 910 Hz |rf| F F |rb| 0.6 Atom 1 'I Atom 2 1000 Hz 'I 0.4 |r| F F 'I 0.55S 'I 1.45S 0.2 910 Hz 0.0 forward reflection amplitude |rf| 0.0 0.5 1.0 1.5 2.0 (b) 'IS 1200 1.50 (d) fout 1090 Hz 1.6 1100 fout (Hz) 1.05 1.2 1000 'I 0.51S 0.8 |r| 0.60 900 'I 1.49S 0.4 0.15 800 0.0 0.0 0.4 0.8 1.2 1.6 2.0 0.0 0.5 1.0 1.5 2.0 'IS 'IS Fig. 3 Unidirectional amplification in a non-Hermitian space−time varying metamaterial. a The “energy-level” diagram for two time-varying meta-atoms with the same resonance frequency (yellow bars) and modulation frequency, and a modulation phase delay Δϕ is applied on atom 2. b The 2D map of the r against the modulation phase delay Δϕ and the output frequency, where arrows indicate the destructive dip position calculated from the harmonic f model. The forward and backward reflection amplitudes spectra with different interference conditions reveal the asymmetric scattering property for both m = −1 (c) and m = 1 harmonics (d). Particularly, unidirectional amplification occurs around Δϕ ¼ 1:4π for the harmonic with m = 1. Here symbols represent the experimental results, and dashed lines represent the numerical simulation results. (a) a(t) (b) T 1180 Hz L F2 3F1 Atom 1 G t 'I Atom 2 'I 1000 Hz Atom 1 F1 910 Hz F2 't L L Atom 2 t F1 G G 820 Hz (c) fin 910 Hz (d) fin 1000 Hz 1.0 1.0 fout 820 Hz fout 820 Hz 0.8 fout 910 Hz 0.8 fout 910 Hz fout 1000 Hz fout 1000 Hz 0.6 fout 1180 Hz 0.6 fout 1180 Hz |tb| |tf| 0.4 0.4 0.2 0.2 0.0 0.0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 'IS 'IS Fig. 4 Analogy of quantum interference with design capability. a The schematic of amplitude modulation for two time-varying meta-atoms, where atom 2 has doubled modulation frequency (F2 ¼ 2 F1 ¼ 180 Hz) and an additional modulation time delay Δt. b The “energy-level” diagram for two time-varying meta-atoms with asymmetric frequency coupling. c Forward transmission amplitude jtf j spectra for input frequency at 910 Hz. d Backward transmission amplitude jtb j for input frequency at 1 kHz, with the interference “turned off”. In c and d, symbols represent the experimental results, dashed lines represent the numerical simulation results, and arrows in c indicate the dip positions predicted from destructive interference conditions (see details in Supplementary Note 5). Analogy of quantum interference with design capability. The flexible control on these interference pathways with tailor-made interference between different pathways in the “energy-level” energy levels. Figure 4a shows a representative example that atom diagram mimics quantum interference, which is the fundamental 2 has doubled modulation frequency (F2 ¼ 2F1 ¼ 180 Hz), an building block in many quantum phenomena, such as electro- additional modulation phase delay induced by the time delay Δt, magnetically induced transparency51. Our system allows very defined as Δϕ ¼ 2πF1 Δt. For this configuration, two meta-atoms COMMUNICATIONS PHYSICS | (2022)5:18 | https://doi.org/10.1038/s42005-021-00790-2 | www.nature.com/commsphys 5
ARTICLE COMMUNICATIONS PHYSICS | https://doi.org/10.1038/s42005-021-00790-2 have different resonating frequencies, as shown in the “energy holes is 0.02 m, the distance ‘ between two meta-atoms is 0.04 m (less than λ=8). level” diagram in Fig. 4b. Here we take the frequency conversion Each meta-atom consists of one speaker (ESE Audio Speaker, with a size of 15 mm ´ 25 mm) and one microphone (Sound Sensor V2 Waveshare, with a size of process from 910 Hz to 1180 Hz as an example to illustrate the 18:7 mm ´ 34:4 mm) interconnecting by a microprocessor (Adafruit ItsyBitsy M4 interference: an up-conversion of m = 3 at atom 1 (red arrow) is Express, with a size of 17:8 mm ´ 35:8 mm) for digital feedback. For the digital competing with an up-conversion of m = 1 at atom 1 and a convolution, the microprocessor is programmed to operate at a sampling frequency successive up-conversion of m = 1 at atom 2 (two gray arrows). of 7.2 kHz (with sampling period Ts = 138.7 μs). For the implementation, a time Two different pathways to the same level interference with each domain convolution kernel yðt Þ ¼ g sinð2πfres ðt δtprog ÞÞe2πγðtδtprog Þ for t > δtprog other, and such an interference can be controlled by varying Δϕ, and 0 for t
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Nonreciprocal gain in non-Hermitian time- material in this article are included in the article’s Creative Commons license, unless Floquet systems. Phys. Rev. Lett. 120, 087401 (2018). indicated otherwise in a credit line to the material. If material is not included in the 43. Song, A. Y., Shi, Y., Lin, Q. & Fan, S. Direction-dependent parity-time phase article’s Creative Commons license and your intended use is not permitted by statutory transition and nonreciprocal amplification with dynamic gain-loss regulation or exceeds the permitted use, you will need to obtain permission directly from modulation. Phys. Rev. A 99, 013824 (2019). the copyright holder. To view a copy of this license, visit http://creativecommons.org/ 44. Chitsazi, M., Li, H., Ellis, F. M. & Kottos, T. Experimental realization of licenses/by/4.0/. Floquet PT-symmetric systems. Phys. Rev. Lett. 119, 093901 (2017). 45. Li, M., Ni, X., Weiner, M., Alù, A. & Khanikaev, A. B. 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